= Fixed-generator commutator collection in nilpotent groups
{title2=$G\prime=[G,a_1]\cdots[G,a_d]$}
If a <nilpotent group> is generated by $a_1,\ldots,a_d$, every element of its <commutator subgroup> is a product $[x_1,a_1]\cdots[x_d,a_d]$. The sets in the displayed formula need not be <subgroups>. Induct on the nilpotency class $c$. Collection in $G/\gamma_c(G)$ gives the desired product with a central error. Since $\gamma_c(G)=[\gamma_{c-1}(G),G]$, that error is a product $[y_1,a_1]\cdots[y_d,a_d]$ with $y_i\in\gamma_{c-1}(G)$. These factors are central, and $\gamma_{c-1}(G)$ centralizes $G\prime$, so replacing $x_i$ by $x_iy_i$ absorbs the error. For a finitely generated <pro-p group>, applying this formula in all finite quotients and using compactness makes its algebraically generated <commutator subgroup> closed.
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